Common Fixed Point Theorems for Weakly Compatible Mappings in Fuzzy Metric Spaces Using (JCLR) Property ()
1. Introduction
In 1965, Zadeh [1] investigated the concept of a fuzzy set in his seminal paper. In the last two decades there has been a tremendous development and growth in fuzzy mathematics. The concept of fuzzy metric space was introduced by Kramosil and Michalek [2] in 1975, which opened an avenue for further development of analysis in such spaces. Further, George and Veeramani [3] modified the concept of fuzzy metric space introduced by Kramosil and Michalek [2] with a view to obtain a Hausdoroff topology which has very important applications in quantum particle physics, particularly in connection with both string and
theory (see, [4] and references mentioned therein). Fuzzy set theory also has applications in applied sciences such as neural network theory, stability theory, mathematical programming, modeling theory, engineering sciences, medical sciences (medical genetics, nervous system), image processing, control theory, communication etc.
In 2002, Aamri and El-Moutawakil [5] defined the notion of (E.A) property for self mappings which contained the class of non-compatible mappings in metric spaces. It was pointed out that (E.A) property allows replacing the completeness requirement of the space with a more natural condition of closedness of the range as well as relaxes the complexness of the whole space, continuity of one or more mappings and containment of the range of one mapping into the range of other which is utilized to construct the sequence of joint iterates. Subsequently, there are a number of results proved for contraction mappings satisfying (E.A) property in fuzzy metric spaces (see [6-11]). Most recently, Sintunavarat and Kumam [12] defined the notion of “common limit in the range” property (or (CLR) property) in fuzzy metric spaces and improved the results of Mihet [10]. In [12], it is observed that the notion of (CLR) property never requires the condition of the closedness of the subspace while (E.A) property requires this condition for the existence of the fixed point (also see [13]). Many authors have proved common fixed point theorems in fuzzy metric spaces for different contractive conditions. For details, we refer to [14-25].
The aim of this paper is to introduce the notion of the joint common limit in the range of mappings property called (JCLR) property and prove a common fixed point theorem for a pair of weakly compatible mappings using (JCLR) property in fuzzy metric space. As an application to our main result, we present a common fixed point theorem for two finite families of self mappings in fuzzy metric space using the notion of pairwise commuting due to Imdad et al. [15]. Our results improve and generalize the results of Cho et al. [26], Abbas et al. [7] and Kumar [8].
2. Preliminaries
Definition 2.1 [27] A binary operation
is a continuous t-norm if it satisfies the following conditions:
1)
is associative and commutative
2)
is continuous
3)
for all
4)
whenever
and
for all
.
Examples of continuous t-norms are
and
.
Definition 2.2 [3] A 3-tuple
is said to be a fuzzy metric space if X is an arbitrary set, * is a continuous t-norm and M is a fuzzy set on
satisfying the following conditions: For all
,
,
1)
2)
if and only if
3)
4)
5)
is continuous.
Then M is called a fuzzy metric on X and
denotes the degree of nearness between x and y with respect to t.
Let
be a fuzzy metric space. For
, the open ball
with center
and radius
is defined by

Now let
be a fuzzy metric space and
the set of all
with
if and only if there exist
and
such that
. Then
is a topology on X induced by the fuzzy metric M.
In the following example (see [3]), we know that every metric induces a fuzzy metric:
Example 2.1 Let
be a metric space. Denote
(or
) for all
and let
be fuzzy sets on
defined as follows:
.
Then
is a fuzzy metric space and the fuzzy metric M induced by the metric d is often referred to as the standard fuzzy metric.
Definition 2.3 Let
be a fuzzy metric space. M is said to be continuous on
if

whenever a sequence
in
converge to a point
, i.e.,

and

Lemma 2.1 [28] Let
be a fuzzy metric space. Then
is non-decreasing for all
.
Lemma 2.2 [29] Let
be a fuzzy metric space. If there exists
such that

for all
and
, then
.
Definition 2.4 [30] Two self mappings f and g of a non-empty set X are said to be weakly compatible (or coincidentally commuting) if they commute at their coincidence points, i.e. if
some
, then
.
Remark 2.1 [30] Two compatible self mappings are weakly compatible, but the converse is not true. Therefore the concept of weak compatibility is more general than that of compatibility.
Definition 2.5 [7] A pair of self mappings f and g of a fuzzy metric space
are said to satisfy the (E.A) property, if there exists a sequence
in X for some
such that

Remark 2.2 It is noted that weak compatibility and (E.A) property are independent to each other (see [31], Example 2.1, Example 2.2).
In 2011, Sintunavarat and Kumam [12] defined the notion of “common limit in the range” property in fuzzy metric space as follows:
Definition 2.6 A pair
of self mappings of a fuzzy metric space
is said to satisfy the “common limit in the range of g” property (shortly, (CLRg) property) if there exists a sequence
in X such that

for some
.
Now, we show examples of self mappings f and g which are satisfying the (CLRg) property.
Example 2.2 Let
be a fuzzy metric space with
and

for all
. Define self mappings f and g on X by
and
for all
. Let a sequence
in X, we have

which shows that f and g satisfy the (CLRg) property.
Example 2.3 The conclusion of Example 2.2 remains true if the self mappings f and g is defined on X by
and
for all
. Let a sequence
in X. Since

therefore f and g satisfy the (CLRg) property.
The following definition is on the lines due to Imdad et al. [32].
Definition 2.7 [32] Two families of self mappings
and
are said to be pairwise commuting if
1)
for all
2)
for all
3)
for all
and
.
Throughout this paper,
is considered to be a fuzzy metric space with condition
for all
.
3. Main Results
In this section, we first introduce the notion of “the joint common limit in the range property” of two pairs of self mappings.
Definition 3.1 Let
be a fuzzy metric space and
. The pair
and
are said to satisfy the “joint common limit in the range of b and g” property (shortly, (JCLRbg) property) if there exists a sequence
and
in X such that
(1)
for some
.
Remark 3.1 If
,
and
in (1), then we get the definition of (CLRg).
Throughout this section,
denotes the set of all continuous and increasing functions
in any coordinate and
for all
.
Following are examples of some function
:
1)
for some
.
2)
for some
.
3)
for some
and for all t-norm
such that
.
Now, we state and prove main results in this paper.
Theorem 3.1 Let
be a fuzzy metric space, where
is a continuous t-norm and f, g, a and b be mappings from X into itself. Further, let the pair
and
are weakly compatible and there exists a constant
such that
(2)
holds for all
,
,
and
. If
and
satisfy the (JCLRbg) property, then f, g, a and b have a unique common fixed point in X.
Proof. Since the pairs
and
satisfy the (JCLRbg) property, there exists a sequence
and
in X such that

for some
.
Now we assert that
. Using (2), with
,
, for
, we get

Taking the limit as
, we have

Since
is increasing in each of its coordinate and
for all
, we get
. By Lemma 2.2, we have
.
Next we show that
. Using (2), with
,
, for
, we get

Taking the limit as
, we have

Since
is increasing in each of its coordinate and
for all
, we get
. By Lemma 2.2, we have
.
Now, we assume that
. Since the pair
is weakly compatible,
and then
. It follows from
is weakly compatible,
and hence
.
We show that
. To prove this, using (2) with
,
, for
, we get

and so

Since
is increasing in each of its coordinate and
for all
,
, which implies that
. Hence
.
Next, we show that
. To prove this, using (2) with
,
, for
, we get

and so

Since
is increasing in each of its coordinate and
for all
,
, which implies that
. Hence
. Therefore, we conclude that
this implies f, g, a and b have common fixed point that is a point z.
For uniqueness of common fixed point, we let w be another common fixed point of the mappings f, g, a and b. On using (2) with
,
, for
, we have

and then

Since
is increasing in each of its coordinate and
for all
,
, which implies that
. Therefore f, g, a and b have a unique a common fixed point.
Remark 3.2 From the result, it is asserted that (JCLRgb) property never requires any condition closedness of the subspace, continuity of one or more mappings and containment of ranges amongst involved mappings.
Remark 3.3 Theorem 3.1 improves and generalizes the results of Abbas et al. ([7], Theorem 2.1) and Kumar ([8], Theorem 2.3) without any requirement of containment amongst range sets of the involved mappings and closedness of the underlying subspace.
Remark 3.4 Since the condition of t-norm with
for all
is replaced by arbitrary continuous t-norm, Theorem 3.1 also improves the result of Cho et al. ([26], Theorem 3.1) without any requirement of completeness of the whole space, continuity of one or more mappings and containment of ranges amongst involved mappings.
Corollary 3.1 Let
be a fuzzy metric space, where
is a continuous t-norm and f, g, a and b be mappings from X into itself. Further, let the pair
and
are weakly compatible and there exists a constant
such that
(3)
holds for all
,
,
and
such that
. If
and
satisfy the (JCLRbg) property, then f, g, a and b have a unique common fixed point in X.
Proof. By Theorem 3.1, if we define

then the result follows.
Remark 3.5 Corollary 3.1 improves the result of Cho et al. ([26], Corollary 3.4) without any requirement of completeness of the whole space, continuity of one or more mappings and containment of ranges amongst involved mappings while the condition of t-norm
for all
is replaced by arbitrary continuous t-norm.
Corollary 3.2 Let
be a fuzzy metric space, where
is a continuous t-norm and f and g be mappings from X into itself. Further, let the pair
is weakly compatible and there exists a constant
such that
(4)
holds for all
,
,
and
. If
satisfies the (CLRg) property, then f and g have a unique common fixed point in X.
Proof. Take
and
in Theorem 3.1, then we get the result.
Our next theorem is proved for a pair of weakly compatible mappings in fuzzy metric space
using (E.A) property under additional condition closedness of the subspace.
Theorem 3.2 Let
be a fuzzy metric space, where
is a continuous t-norm. Further, let the pair
of self mappings is weakly compatible satisfying inequality (4) of Corollary 3.2. If f and g satisfy the (E.A) property and the range of g is a closed subspace of X, then f and g have a unique common fixed point in X.
Proof. Since the pair
satisfies the (E.A) property, there exists a sequence
in X such that

for some
. It follows from
being a closed subspace of X that there exists
in which
. Therefore f and g satisfy the (CLRg) property. From Corollary 3.2, the result follows.
In what follows, we present some illustrative examples which demonstrate the validity of the hypotheses and degree of utility of our results.
Example 3.1 Let
with the metric d defined by
and for each
define

for all
. Clearly
be a fuzzy metric space with t-norm defined by
for all
. Consider a function
defined by
. Then we have
. Define the self mappings f and g on X by

and

Taking
or
, it is clear that the pair
satisfies the (CLRg) property since

It is noted that
. Thus, all the conditions of Corollary 3.2 are satisfied for a fixed constant
and 2 is a unique common fixed point of the pair
. Also, all the involved mappings are even discontinuous at their unique common fixed point 2. Here, it may be pointed out that
is not a closed subspace of X.
Example 3.2 In the setting of Example 3.1, replace the mapping g by the following, besides retaining the rest:

Taking
or
, it is clear that the pair
satisfies the (E.A) property since

It is noted that
. Thus, all the conditions of Theorem 3.2 are satisfied and 2 is a unique common fixed point of the mappings f and g. Notice that all the involved mappings are even discontinuous at their unique common fixed point 2. Here, it is worth noting that
is a closed subspace of X.
Now, we utilize Definition 2.7 which is a natural extension of commutativity condition to two finite families of self mappings. Our next theorem extends Corollary 3.2 in the following sense:
Theorem 3.3 Let
and
be two finite families of self mappings in fuzzy metric space
, where
is a continuous t-norm such that
and
which satisfy the inequalities (4) of Corollary 3.2. If the pair
shares (CLRg) property, then f and g have a unique point of coincidence.
Moreover,
and
have a unique common fixed point provided the pair of families 
commutes pairwise, where
and
.
Proof. The proof of this theorem can be completed on the lines of Theorem 3.1 contained in Imdad et al. [15], hence details are avoided.
Putting
and
in Theorem 3.3, we get the following result:
Corollary 3.3 Let f and g be two self mappings of a fuzzy metric space
, where
is a continuous t-norm. Further, let the pair
shares (CLRg) property. Then there exists a constant
such that

holds for all
,
,
,
and m and n are fixed positive integers, then f and g have a unique common fixed point provided the pair
commutes pairwise.
Remark 3.6 Theorem 3.2, Theorem 3.3 and Corollary 3.3 can also be outlined in respect of Corollary 3.1.
Remark 3.7 Using Example 2.2, we can obtain several fixed point theorems in fuzzy metric spaces in respect of Theorems 3.2 and 3.3 and Corollaries 3.2, 3.1 and 3.3.
4. Acknowledgements
The authors would like to express their sincere thanks to Professor Mujahid Abbas for his paper [18]. The second author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST). This study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission under the Computational Science and Engineering Research Cluster (CSEC Grant No. 55000613).
NOTES
#Corresponding authors.